This book on Integration Theory is based on the lecture notes for courses that the author gave at Tata Institute of Fundamental Research, Mumbai and at ETH, Zurich. The subject matter is classical. The goal of the notes is to provide a concise, clear and accurate treatment of the basic ideas of the subject.
1. Integration on a measure space 2. The Lebesgue spaces 3. The outer measure and its applications: the Lebesgue measure 4. Product measures and multiple integrals 5. Set functions and their derivatives.
As the author asserts, the material presented in this slim volume is classical; his goal has been "concision, clarity, and accuracy".
The author is extraordinarily careful in detail, for example in showing that the integrals of simple functions and integrable functions are well-defined.
Texts and Readings in Mathematics/ 8
2011 126 pages paper cover ISBN 978-93-80250-19-9
This book contains the author's notes for a course by him at ETH, Zurich. The aim is to lead the reader to a proof of the Peter-Weyl Theorem, the basic theorem in the representation theory of compact topological groups. The topological, analytical, and algebraic groundwork needed for the proof is provided as part of the course.
1. Topological Preliminaries 2. The Haar measure on a locally compact group 3. Hilbert spaces and the spectral theorem 4. Compact groups and their representations.
The presentation in this text is clear and to the point. The methods used are good classical ones.
This is a good text for a student who knows little about locally compact groups and wants to get an introduction to some of the fundamental ideas needed to begin the study of them.
Texts and Readings in Mathematics/ 9
2011 126pages paper cover ISBN 978-93-80250-20-5
The purpose of this book is multifold. The FIRST and foremost is to give an elementary introduction to the basic concepts of the theory of ordinary representations to finite groups with a minimum of prerequisites.
The SECOND which is also the main theme of this exposition is to be able to do the theory rather explicitly for the important special case of the symmetric groups Sn of permutations on n letters.
The THIRD aspect is to use the preparatory material of the first two parts coupled with the Sn theory to do the same for some other important special groups, namely, the alternating group An the Hyperoctahedral groups Bn and Dn.
Part I : The Structure of Semi-simple Rings. 1. Preliminaries. 2. Semi-simple Rings and Brauer Group.
Part II. Representations of Finite Groups. 3. Representations of Finite Groups. 4. Induced Representations.
Part III. Representations of the Symmetric and Alternating Groups. 5. Representations of the Symmetric Group Sn. 6. Representations of the Alternating group An. Part IV. Representations of the Hyperoctahedral Groups Bn and Dn . 7. epresentations of the Hyperoctahedral Group Bn 8. Representations of the Hyperoctahedral Group Dn.
The book under review presents an interesting approach to certain aspects of the representation theory of finite groups over the complex field.
Texts and Readings in Mathematics/ 3
2011 250 pages paper cover ISBN 978-93-80250-18-2
This book treats some basic topics in the spectral theory of Dynamical Systems. The treatment is at a general level, but even here, two theorems which are not on the surface, one due to H. Helson and W. Parry, and the other due to B. Host, are presented. Moreover, Ornstein's family of mixing rank one automorphisms is described with construction and proof. Systems of imprimitivity and their relevance for Ergodic Theory is discussed, and Baire category theorems of Ergodic Theory, scattered in the literature, are derived in a unified way. Riesz products are considered and they are used to describe the spectral types and eigenvalues of rank one automorphisms.
1. The Hahn-Hellinger Theorem 2. The Spectral Theorem for Unitary Operators 3. Symmetry and Denseness of the Spectrum 4. Multiplicity and Rank 5. The Skew Product 6. A Theorem of Helson and Parry 7. Probability Measures on the Circle Group 8. Baire Category Theorems of Ergodic Theory 9. Translations of Measures on the Circle 10. B. Host?s Theorem 11. L Eigenvalues of Non-Singular Automorphisms 12. Generalities on Systems of Imprimitivity 13. Dual Systems of Imprimitivity 14. Saturated Subgroups of the Circle Group 15. Riesz Products as Spectral Measures 16. Additional Topics.
ate in a unique way a lot of mathematics that is of current interest but is not otherwise so readily at hand.
Texts and Readings in Mathematics/ 15
2011 226 pages paper cover ISBN 978-93-80250-21-2
(I.H.E.S.-PUB. MATH. VOL.113) 2011
- Juha HEINONEN and Stephen KEITH
Flat forms, bi-lipschitz parametrizations, and smoothability of manifolds
p. 1-37
- Rupert L. FRANK, Elliott H. LIEB, Robert SEIRINGER, and Lawrence E. THOMAS
Stability and absence of binding for multi-polaron systems
p. 39-67
- Damien GAYET and Jean-Yves WELSCHINGER
Exponential rarefaction of real curves with many components
p. 69-96
- P.B. KRONHEIMER and T.S. MROWKA
Khovanov cohomology is an unknot-detector
p. 97-208
(I.H.E.S.-PUB. MATH. VOL.112) 2010
- Vladimir VOEVODSKY
Motivic Eilenberg-MacLane spaces
p. 1-99
- Pierre DELIGNE
Le Groupe fondamental unipotent motivique de Gm - ƒÊN pour N = 2, 4, 6 ou 8
p. 101-141
- A. ALEKSEEV, B. ENRIQUEZ and C. TOROSSIAN
Drinfeld associators, braid groups and explicit solutions of the Kashiwara-Vergne equations
p. 143-189
- Mohammed ABOUZAID
A Geometric criterion for generating the Fukaya category
p. 191-240
(I.H.E.S.-PUB. MATH. VOL.111) 2010
- BAO CHAU Ngo
Le lemme fondamental pour les algebres de Lie-
p. 1-169
- Philippe MICHEL
The subconvexity problem for GL2
p. 171-271